Constant Rank-Distance Sets of Hermitian Matrices and Partial Spreads in Hermitian Polar Spaces

In this paper we investigate partial spreads of H(2n− 1, q2) through the related notion of partial spread sets of hermitian matrices, and the more general notion of constant rank-distance sets. We prove a tight upper bound on the maximum size of a linear constant rank-distance set of hermitian matrices over finite fields, and as a consequence prove the maximality of extensions of symplectic semifield spreads as partial spreads of H(2n−1, q2). We prove upper bounds for constant rank-distance sets for even rank, construct large examples of these, and construct maximal partial spreads of H(3, q2) for a range of sizes.